Probabilistic Galois Theory

Abstract

We show that there are at most $O_{n,\epsilon }(H^{n-2+\sqrt{2}+\epsilon })$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_n(H^{n-1/2}\log H)$.

10 pages

Cited by in corpus (5)

• Enumerative Galois theory for cubics and quartics
• Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants
• Distribution of Frobenius elements in families of Galois extensions
• Splitting Behavior of $S_n$-Polynomials
• Irreducibility of the characteristic polynomials of random tridiagonal matrices